—. — , - 0^ , and + (9„ . Thus, in order to find the beam parameters, all that is 



dx d\ d\ ^ 



is needed from the theory of elasticity is the distribution of the stresses over the cross 



section! 



These equations can be simplified somewhat by choosing a particular coordinate 



system. Let 



ros 



J T? 



\; = / ^r^dA 



Then 



and 



12 ~ ^21 ' ®^^- 



5U, 

 -^ =IiiV. +I12M, +I13M. 



^ 21x 22y li z 



^=l3lV.+l32M, +I33M. 



Choose a new coordinate system whose origin is at y, z in the original coordinate system. 

 Let Ujj, 6^ , 6^ , Vjj, M , and M^ be the unknowns in this new system (see Figure 3). Then 



U = U +^ 6 -yd 



X X y •' z 



^y = ^y; ^z=^z 



V, =V,; M =M +zV,; M^=M^-7v, 



*To similarly derive expressions for dO /d\, ((9u /(9x ) - 6^, {d\J /d\ ) + 6 in terms of assumed distribu- 

 tions of a^ and a^^ over the cross section we would repeat the development for dU /dx , dd /dx, dd / dx 

 almost identically. The importance of this derivation is not the integrals (GjG2/E)dA, etc., but the demonstra- 

 tion that the strain energy per unit length W is a quadratic form in the terms V , M , M (for terms dependent 

 on C'j^x^^"'^ ^ quadratic form in the terms V , V , M (for terms dependent on and O" ), see page 74. 



The value of the expressions for dU /dx , etc., is to validate the use of Castigliano's Theorem in obtaining 

 expressions for flexibility terms based on energy expressions. 



rts seen in the subsequent theory, the actual distribution chosen for a and a (i.e., the shear flows in the 

 plates) is forced to be compatible with and dependent on the distribution chosen for cr ; namely, that indicated 

 by Fj = 1, F2 = y, F3 = z, and cr^^(.y, z) = KjFj + K2F2 + K3F,, which is restated on page 52. 



48 - - 



